WKB analysis of the Logarithmic Nonlinear Schrodinger Equation in an analytic framework

TL;DR

This paper develops a WKB analysis for the logarithmic nonlinear Schrödinger equation within an analytic framework, establishing local well-posedness and the semiclassical limit, applicable to a broad class of initial data.

Contribution

It introduces a novel analytic framework for WKB analysis of the logarithmic NLS, accommodating initial data that vanish at infinity and ensuring uniform well-posedness and semiclassical convergence.

Findings

Established local well-posedness uniformly in the semiclassical parameter

Demonstrated the validity of the semiclassical limit in the analytic setting

Extended applicability to initial data converging to zero at infinity

Abstract

We are interested in a WKB analysis of the Logarithmic Non-Linear Schr\"odinger Equation with "Riemann-like" variables in an analytic framework in semiclassical regime. We show that the Cauchy problem is locally well posed uniformly in the semiclassical constant and that the semiclassical limit can be performed. In particular, our framework is not only compatible with the Gross-Pitaevskii equation with logarithmic nonlinearity, but also allows initial data (and solutions) which can converge to $0$ at infinity.